Differential Equations With Mathematica, Fifth Edition
Martha L. Abell and James P. Braselton
Contents
Preface ix
Introduction to differential equations
1.1 Definitions and concepts 1
1.2 Solutions of differential equations 4
1.3 Initial- and boundary-value problems 12
1.4 Direction fields 18
1.4.1 Creating interactive applications 27
2. First-order ordinary differential equations
2.1 Theory of first-order equations: a brief discussion 29
2.2 Separation of variables 33
2.3 Homogeneous equations 42
2.4 Exact equations 46
2.5 Linear equations 50
2.5.1 Integrating factor approach 51
2.5.2 Variation of parameters and the method of undetermined coefficients 56
2.6 Numerical approximations
of solutions to first-order
equations 60
2.6.1 Built-in methods 60
2.6.2 Other numerical
methods 63
Application: modeling
the spread of a disease 77
Applications of
first-order equations
3.1 Orthogonal trajectories 83
3.2 Population growth and
decay 90
3.2.1 The Malthus model 90
3.2.2 The logistic equation 95
3.3 Newton’s law of cooling 104
3.4 Free-falling bodies 108
Higher-order linear
differential equations
4.1 Preliminary definitions and
notation 115
4.1.1 Introduction 115
4.1.2 The nth-order
ordinary linear
differential equation 119
4.1.3 Fundamental set of
solutions 124
4.1.4 Existence of a
fundamental set of
solutions 127
4.1.5 Reduction of order 128
4.2 Solving homogeneous
equations with constant
coefficients 131
4.2.1 Second-order
equations 131
4.2.2 Higher-order
equations 135
4.3 Introduction to solving
nonhomogeneous equations 141
4.4 Nonhomogeneous
equations with constant
coefficients: the method of
undetermined coefficients 145
4.4.1 Second-order
equations 147
4.4.2 Higher-order
equations 158
4.5 Nonhomogeneous equations with constant coefficients: variation of parameters 164
4.5.1 Second-order
equations 164
4.5.2 Higher-order
nonhomogeneous
equations 167
4.6 Cauchy–Euler equations 170
4.6.1 Second-order
Cauchy–Euler
equations 170
4.6.2 Higher-order
Cauchy–Euler
equations 173
4.6.3 Variation of
parameters 177
4.7 Series solutions 179
4.7.1 Power series solutions
about ordinary points 179
4.7.2 Series solutions about
regular singular points 189
4.7.3 Method of Frobenius 190
Application: zeros of
the Bessel functions of
the first kind 200
4.8 Nonlinear equations 205
Applications of
higher-order differential
equations
5.1 Harmonic motion 221
5.1.1 Simple harmonic motion 221
5.1.2 Damped motion 228
5.1.3 Forced motion 238
5.1.4 Soft springs 250
5.1.5 Hard springs 253
5.1.6 Aging springs 254
Application: hearing
beats and resonance 255
5.2 The pendulum problem 256
5.3 Other applications 265
5.3.1 L-R-C circuits 265
5.3.2 Deflection of a beam 268
5.3.3 Bodé plots 270
5.3.4 The catenary 273
Systems of ordinary
differential equations
6.1 Review of matrix algebra
and calculus 283
6.1.1 Defining nested lists,
matrices, and vectors 283
6.1.2 Extracting elements of
matrices 287
6.1.3 Basic computations
with matrices 288
6.1.4 Systems of linear
equations 290
6.1.5 Eigenvalues and
eigenvectors 292
6.1.6 Matrix calculus 296
6.2 Systems of equations:
preliminary definitions and
theory 297
6.2.1 Preliminary theory 301
6.2.2 Linear systems 308
6.3 Homogeneous linear
systems with constant
coefficients 315
6.3.1 Distinct real
eigenvalues 315
6.3.2 Complex conjugate
eigenvalues 320
6.3.3 Solving initial-value
problems 325
6.3.4 Repeated eigenvalues 327
6.4 Nonhomogeneous
first-order systems:
undetermined coefficients,
variation of parameters, and
the matrix exponential 333
6.4.1 Undetermined
coefficients 334
6.4.2 Variation of
parameters 337
6.4.3 The matrix
exponential 342
6.5 Numerical methods 348
6.5.1 Built-in methods 349
6.5.2 Euler’s method 356
6.5.3 Runge–Kutta method 360
6.6 Nonlinear systems,
linearization, and
classification of equilibrium
points 362
6.6.1 Real distinct
eigenvalues 363
6.6.2 Repeated eigenvalues 368
6.6.3 Complex conjugate
eigenvalues 371
6.6.4 Nonlinear systems 374
Applications of systems
of ordinary differential
equations
7.1 Mechanical and electrical
problems with first-order
linear systems 385
7.1.1 L-R-C circuits with
loops 385
7.1.2 L-R-C circuit with
one loop 385
7.1.3 L-R-C circuit with
two loops 387
7.1.4 Spring–mass systems 390
7.2 Diffusion and population
problems with first-order
linear systems 391
7.2.1 Diffusion through a
membrane 391
7.2.2 Diffusion through a
double-walled
membrane 393
7.2.3 Population problems 396
7.3 Applications that lead to
nonlinear systems 399
7.3.1 Biological systems:
predator–prey
interactions, the
Lotka–Volterra system,
and food chains in the
chemostat 400
7.3.2 Physical systems:
variable damping 414
7.3.3 Differential geometry:
curvature 418
Laplace transform
methods
8.1 The Laplace transform 423
8.1.1 Definition of the
Laplace transform 423
8.1.2 Exponential order,
jump discontinuities,
and piecewise
continuous functions 426
8.1.3 Properties of the
Laplace transform 428
8.2 The inverse Laplace
transform 432
8.2.1 Definition of the
inverse Laplace
transform 432
8.2.2 Laplace transform of
an integral 438
8.3 Solving initial-value
problems with the Laplace
transform 439
8.4 Laplace transforms of step
and periodic functions 444
8.4.1 Piecewise defined
functions: the unit
step function 444
8.4.2 Solving initial-value
problems with
piecewise continuous
forcing functions 447
8.4.3 Periodic functions 449
8.4.4 Impulse functions: the
delta function 457
8.5 The convolution theorem 462
8.5.1 The convolution
theorem 462
8.5.2 Integral and
integrodifferential
equations 464
8.6 Applications of Laplace
transforms, Part I 466
8.6.1 Spring–mass systems
revisited 466
8.6.2 L-R-C circuits
revisited 469
8.6.3 Population problems
revisited 474
Application: the
tautochrone 476
8.7 Laplace transform methods
for systems 478
8.8 Applications of Laplace
transforms, Part II 488
8.8.1 Coupled spring–mass
systems 488
8.8.2 The double pendulum 493
Application: free
vibration of a
three-story building 496
Eigenvalue problems and
Fourier series
9.1 Boundary-value problems,
eigenvalue problems, and
Sturm–Liouville problems 503
9.1.1 Boundary-value
problems 503
9.1.2 Eigenvalue problems 505
9.1.3 Sturm–Liouville
problems 508
9.2 Fourier sine series and
cosine series 510
9.2.1 Fourier sine series 510
9.2.2 Fourier cosine series 515
9.3 Fourier series 518
9.3.1 Fourier series 518
9.3.2 Even, odd, and
periodic extensions 525
9.3.3 Differentiation and
integration of Fourier
series 530
9.3.4 Parseval’s equality 534
9.4 Generalized Fourier series 535
10.Partial differential
equations
10.1 Introduction to partial
differential equations and
separation of variables 545
10.1.1 Introduction 545
10.1.2 Separation of
variables 546
10.2 The one-dimensional heat
equation 548
10.2.1 The heat equation
with homogeneous
boundary conditions 548
10.2.2 Nonhomogeneous
boundary conditions 551
10.2.3 Insulated boundary 554
10.3 The one-dimensional wave
equation 557
10.3.1 The wave equation 557
10.3.2 D’Alembert’s solution 562
10.4 Problems in two dimensions:
Laplace’s equation 565
10.4.1 Laplace’s equation 565
10.5 Two-dimensional problems
in a circular region 570
10.5.1 Laplace’s equation in
a circular region 570
10.5.2 The wave equation in
a circular region 573
Bibliography 585
Index 587